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Programming in Logic: Prolog

Programming in Logic: Prolog. Prolog’s Declarative & Procedural Semantics Readings: Sections 2.3 - 2.7. Where is the Prolog Program?. When one talks about Prolog programs, one is referring to the Knowledge base. The definitions of the relations in the Knowledge base is the Prolog code. Goals.

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Programming in Logic: Prolog

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  1. Programming in Logic: Prolog Prolog’s Declarative & Procedural Semantics Readings: Sections 2.3 - 2.7 CS360 Lecture 3

  2. Where is the Prolog Program? • When one talks about Prolog programs, one is referring to the Knowledge base. • The definitions of the relations in the Knowledge base is the Prolog code. CS360 Lecture 3

  3. Goals • Goals are the terms that Prolog attempts to match against the clause heads in its Knowledge base. • The top level terms in a query are all goals: • ?: a(4, x), b(Y,7). • The top level terms in the body of a clause are all goals: • p(X) :- q(X, a), r(3, X). CS360 Lecture 3

  4. Goals and Data Structures • Prolog goals look exactly like Prolog data structures. • In other words, Prolog code looks just like Prolog data. This is intentional! • A Prolog program can modify itself as it is running. • A Prolog program can create new queries & pose them to the Prolog interpreter. CS360 Lecture 3

  5. Prolog’s Declarative Semantics • The prolog we have seen so far is called pure Prolog, i.e., everything can be expressed in FOPC. • For example: ancestor(A,P) :- parent(A,X), ancestor(X,P). can be translatedinto the logical expression: AP(ancestor(A,P)  parent(A,X)  ancestor(X,P)) CS360 Lecture 3

  6. Translating Prolog clauses into FOPC • The clauses : ancestor(A,P) :- parent(A,P). ancestor (A,P):- parent(A,X), ancestor(X,P). Translates into FOPC as: AP(ancestor(A,P)  parent(A,P)  ( parent(A,X)  ancestor(X,P))) CS360 Lecture 3

  7. Prolog Declarative Semantics • In the body of the clause, the commas separating the top-level terms are understood to be conjunctions (and’s). • There are two ways to specify disjunctions (or’s): • Write separate clauses (as done for ancestor) • Use a semicolon (“;”) CS360 Lecture 3

  8. Disjunctions in Prolog • We could rewrite the ancestor example in Prolog as follows: ancestor(A,P) :- parent(A,P) ; parent(A,X), ancestor(X,P). CS360 Lecture 3

  9. Negation in Pure Prolog • We cannot express negation in the clause body in pure prolog: • For example, we can’t express in pure Prolog the following FOPC expression HW(husband(H,W)  married(H,W)  male(H)  divorced(H,W) ) CS360 Lecture 3

  10. Negation in Pure Prolog cont’d • Likewise, Prolog can’t tell us something is false rather it tells us that it can’t prove something is true. • For example, given the Knowledge base: • man(socrates). • mortal(X) :- man(X). • The answer to mortal(mike) would be no, not because it isn’t true, but because Prolog can’t prove it’s true using that KB. CS360 Lecture 3

  11. Prolog’s Procedural Semantics I • Prolog’s control flow different from languages like Java. • In Prolog no distinction between statements & method calls - all goals recursively matched against KB (system defined relations (write/1) have own definitions). • Goal parameters can be unbound variables. • Goals succeed or they fail. CS360 Lecture 3

  12. Prolog Control Flow Example • Given the query a(2), Prolog scans the KB for the first clause head that matches the first query goal. • If it finds a match then it recurses on the goals in the clause body until it bottoms out. • Query succeeded here. KB 1. a(X) :- b(X). 2. b(2). Execution trace of query [a(2)] (goal stack) 1. X = 2 (which clause [b(2)] matched & bindings) 2. [ ] CS360 Lecture 3

  13. Prolog Control Flow Example • Query: a(2) • Notice that after the query goal matched the clause head, the query goal (a(2)) is replaced by the first clause body goal (b(2)) in the next step of the trace with its variable bound. KB 1. a(X) :- b(X). 2. b(2). Execution trace of query [a(2)] (goal stack) 1. X = 2 (which clause [b(2)] matched & bindings) 2. [ ] CS360 Lecture 3

  14. Prolog Control Flow Example • Query:a(3), Prolog scans the KB for first clause head matching first query goal. • If it fails to find a match then it tries to backtrack its matches. • There are no other possible matches. • Query fails. KB 1. a(X) :- b(X). 2. b(2). Execution trace of query [a(3)] 1. X = 3 [b(3)]  CS360 Lecture 3

  15. Backtracking • Query:a(3), from point of failure, Prolog backtracks to last match (matches clause 1), Prolog continues its scan of KB for another clause head that matches goal. • Matches clause 2 and succeeds. KB : 1. a(X) :- b(X). 2. a(3). 3. b(2). Execution trace of query [a(3)] 1. X = 3 2. [b(3)] [ ]  CS360 Lecture 3

  16. Flow of Control with Conjunctions [a(X)] 1.X = X’ I[b(X’), c(X’)] 2. X’ = 2 [c(2)] 3. [] Note renaming of variables in matched clause, X replaced by X’. Goal query is replaced by first clause body goal and rest of clause body goals are pushed onto goal stack. Query: a(X). KB: 1. a(X) :- b(X), c(X). 2. b(2). 3. c(2). Response: X = 2 ? CS360 Lecture 3

  17. Flow of Control with Conjunctions [a(X)] 1.X = X’ [b(X’),c(X’)] 2. X’ = 2 3. X’ = 3 [c(2)] [c(3)]  4. [ ] Query: a(X). KB: 1. a(X) :- b(X), c(X). 2. b(2). 3. b(3) 4. c(3). Response: X = 3 ? Notice unbinding of variable (X’) when backtrack over binding point. CS360 Lecture 3

  18. Prolog Procedural Semantics II • When goal fails, control undoes any bindings done at that step & backs up to previous step. • It continues scan of KB for clause heads matching current goal. • If none found then continues to backtrack. CS360 Lecture 3

  19. Flow of Control with Conjunctions [a(X)] 1.X = X’ [b(X’), c(X’)] 2. X’ = Y [d(Z), e(Y,Z), c(Y)] 5. Z = 5 [e(Y,5), c(Y)] 6. Y = 3 [c(3)] 4. [ ] Query: a(X). KB: 1. a(X) :- b(X), c(X). 2. b(Y) :- d(Z), e(Y,Z). 3. b(3) 4. c(3). 5. d(5). 6. e(3,5). Response: X = 3 ? CS360 Lecture 3

  20. Flow of Control with Conjunctions [a(X)] 1.X = X’ [b(X’), c(X’)] 2. X’ = Y [d(Z),e(Y,Z), c(Y)] 5. Z = 5 [e(Y, 5), c(Y)] 6. Y = 3 7. Y = 4 [c(3)] [c(4)]  4. [ ] Query: a(X). KB: 1. a(X) :- b(X), c(X). 2. b(Y) :- d(Z), e(Y,Z). 3. b(3) 4. c(4). 5. d(5). 6. e(3,5). 7. e(4,5). Response: X = 4 ? CS360 Lecture 3

  21. Additional Answers • What happens when Prolog returns an answer and prompts you with “?”? • Since an answer was returned, the trace ended with a true and an empty goal stack. • If you type in “;” then Prolog will, in effect, change that true to fails and backtrack. CS360 Lecture 3

  22. Effects of Clause/Goal Ordering • A Prolog program can be declaratively correct, but procedurally inadequate (may recurse forever (p :- p.), or simply be inefficient). • The program’s behavior is determined by the order of the clauses in the KB and the order of the goals in the clauses. • Changing either can dramatically affect the behavior of the program. CS360 Lecture 3

  23. Example: Parental Fact KB • parent(pam,bob). • parent(tom,bob). • parent(tom, liz). • parent(bob, ann). • parent(bob,pat). • parent(pat, jim). CS360 Lecture 3

  24. Predecessor Relation Variations pred1(X,Z) :- parent(X, Z). pred1(X,Z) :- parent(X,Y), pred1(Y,Z). pred2(X,Z) :- parent(X,Y), pred2(Y,Z). pred2(X,Z) :- parent(X, Z). pred3(X,Z) :- parent(X, Z). pred3(X,Z) :- pred3(Y,Z), parent(X,Y). pred4(X,Z) :- pred4(Y,Z), parent(X,Y). pred4(X,Z) :- parent(X, Z). CS360 Lecture 3

  25. Computing predn(tom,pat) • For n from 1 to 3, Prolog returns yes. • For pred4(tom, pat), Prolog loops forever. • While the first 3 preds find an answer they do so with different amounts of computation. CS360 Lecture 3

  26. Trace of pred1(tom,pat) [pred1(tom,pat)] 7. X=tom, Z=pat 8. X=tom,Z=pat [parent(tom,pat)] [parent(tom,Y),pred1(Y,pat)]  2. Y=bob [pred1(bob,pat)] 7. X’=bob, Z’=pat [parent(bob,pat)] 5. [ ] KB: 1. parent(pam,bob). 2. parent(tom,bob). 3. parent(tom, liz). 4. parent(bob, ann). 5. parent(bob,pat). 6. parent(pat, jim). 7. pred1(X,Z) :- parent(X, Z). 8. pred1(X,Z) :- parent(X,Y), pred1(Y,Z). CS360 Lecture 3

  27. Exercises • Try doing the execution traces of pred2(tom,pat), pred3(tom,pat), and pred4(tom,pat) CS360 Lecture 3

  28. Quick Quiz • What is the difference between the structure of Prolog code and Prolog data? • What is a goal and how is it different from structured data? • How do you express a negated goal? • What does it mean when Prolog answers no CS360 Lecture 3

  29. Quick Quiz cont’d • In pure Prolog, does changing the order of the clauses or of the goals in the clauses change the declarative meaning? • What about the procedural meaning? • Is there anything declaratively wrong with: • ancestor(A,P) :- ancestor(X,P), parent(A,X). • ancestor(A,P) :- parent(A,P). • Is there anything procedurally wrong? CS360 Lecture 3

  30. Summary • Declarative semantics of pure Prolog program defines when a goal is true and for what instantiation of variables. • Commas between goals mean and, semicolons mean or. • Procedural semantics are determined by clause and goal ordering. • Goal failure causes backtracking and unbinding of affected variables. CS360 Lecture 3

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